Section 1.8 Homework questions?

Section Concepts 1.8 Linear Equations in Two Variables Slide 2 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Definition of a Linear Equation in Two Variables 2.Graphing Linear Equations in Two Variables by Plotting Points 3.x- and y-Intercepts 4.Horizontal and Vertical Lines

DEFINITIONLinear Equation in Two Variables Slide 3 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Let A, B, and C be real numbers such that A and B are not both zero. Then, an equation that can be written in the form: Ax + By = C is called a linear equation in two variables. A solution of a linear equation in two variables is an ordered pair that makes the equation a true statement.

Example 1Determining Solutions to a Linear Equation Slide 4 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. For the linear equation 4x – 5y = 8, determine whether the given ordered pair is a solution.

Example 2You Try Slide 5 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. For the linear equation 3x – 2y = -12, determine whether the given ordered pair is a solution. a. (4, 0)b. (1, )

Example 3You Try Slide 6 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Is (2, 2) a solution of 3x + 2y = 10? 2.Is (-3, 4) a solution of 3x + 2y = 10?

DEFINITIONThe Graph of an Equation in Two Variables Slide 7 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The graph of an equation in two variables is a drawing of all ordered pair solutions to the equation. For a linear equation in two variables, the graph is a straight line.

Section 1.8 Linear Equations in Two Variables 2.Graphing Linear Equations in Two Variables by Plotting Points Slide 8 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The word linear means relating to or resembling a line. Because two points determine a line, to graph a linear equation it is sufficient to find two solution points and draw the line between them; finding a third point can be used to check accuracy.

PROCEDUREGraphing Linear Equations in the Form y = mx + b 1.Choose any value for x. 2.Plug it in and solve for y. 3.(x, y) is a solution to the equation. 4.Find two more solutions (ordered pairs). 5.Plot the points and draw the line connecting them.

Example 4Graphing a Linear Equation Slide 10 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Graph the equation.

Example 5Graphing a Linear Equation Slide 11 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Graph the equation.

Example 6Graphing a Linear Equation Slide 12 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Graph the equation

DEFINITIONx- and y-Intercepts Slide 13 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. An x-intercept of a graph is a point (a, 0) where the graph intersects the x-axis. A y-intercept of a graph is a point (0, b) where the graph intersects the y-axis.

PROCEDUREGraphing Ax + By = C We can use the x and y intercepts to graph equations where x and y are on the same side of the equation.

PROCEDUREFinding x- and y-Intercepts Slide 15 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 Find the x-intercept(s) by substituting y = 0 into the equation and solving for x. Step 2 Find the y-intercept(s) by substituting x = 0 into the equation and solving for y.

Example 7Graphing a Linear Equation Slide 16 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. a.Find the x-intercept. b.Find the y-intercept. c.Graph the equation.

Example 8Finding the x- and y-Intercepts of a Line Slide 17 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 9Finding the x- and y-Intercepts of a Line Slide 18 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Given the equation 6x – 12y = 24 a.Find the x-intercept. b.Find the y-intercept. c.Graph the equation.

DEFINITIONEquations of Vertical and Horizontal Lines Slide 19 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1. A vertical line can be represented by an equation of the form, x = k, where k is a constant. 2. A horizontal line can be represented by an equation of the form, y = k, where k is a constant. Thinking ahead…. What does the line x = 4 look like? What does the line y = 4 look like? What does the line x = 0 look like? What does the line y = 0 look like?

Example 10Graphing a Horizontal Line Slide 20 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Graph the equation y = 3.

Example 11Graphing a Vertical Line Slide 21 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Graph the equation 4x = –8.

Example 12You Try Slide 22 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Graph y = 2x – 3 2.Graph y = 3.Find the x-intercept, y-intercept of x – 3y = -4 4.Graph 5y = Graph x = 4

Example 12You Try Slide 23 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1. Graph y = 2x – 3

Example 12You Try Slide 24 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2. Graph y =

Example 12You Try Slide 25 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3. Find the x-intercept, y-intercept of x – 3y = -4

Example 12You Try Slide 26 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 4. Graph 5y = -10

Example 12You Try Slide 27 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 5. Graph x = 4

Section Concepts 1.8 Linear Equations in Two Variables Slide 28 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Definition of a Linear Equation in Two Variables 2.Graphing Linear Equations in Two Variables by Plotting Points 3.x- and y-Intercepts 4.Horizontal and Vertical Lines

PROCEDUREGraphing Linear Equations in the Form y = mx + b 1.Choose any value for x. 2.Plug it in and solve for y. 3.(x, y) is a solution to the equation. 4.Find two more solutions (ordered pairs). 5.Plot the points and draw the line connecting them.

PROCEDUREGraphing Ax + By = C We can use the x and y intercepts to graph equations where x and y are on the same side of the equation.

PROCEDUREFinding x- and y-Intercepts Slide 31 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 Find the x-intercept(s) by substituting y = 0 into the equation and solving for x. Step 2 Find the y-intercept(s) by substituting x = 0 into the equation and solving for y.

PROCEDUREEquations of Vertical and Horizontal Lines Slide 32 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1. A vertical line can be represented by an equation of the form, x = k, where k is a constant. 2. A horizontal line can be represented by an equation of the form, y = k, where k is a constant.